Realizing Strongly Quasipositive Links and Bennequin Surfaces

Abstract

Adapting the graph index to be used for braided surfaces, we gain realization results on string numbers of Bennequin surfaces. In particular, all fibered alternating links and pure alternating pretzel links have a minimal string Bennequin surface (as can also be proved for many cables of such knots). We also obtain a visual test for strong quasipositivity, which we call pseudo-positivity, and use it to classify strongly quasipositive pretzel knots and reverse Montesinos links. This test also supports determining the strong quasipositivity status of all so far tabulated (up to 16 crossing prime) knots.